if f(x - 1) + f(x + 1) = √3 f(x)
then prove that f(x) is periodic with period 12`
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1 Answers
1057we have
f(x+1)+f(x-1)=√3f(x) (i)
replacing x by x-1 and x+1 in eq.(i),then
f(x)+f(x-2)=√3f(x-1) (ii)
f(x+2)+f(x)=√3f(x+1) (iii)
adding (ii) and (iii) we get
2f(x)+f(x-2)+f(x+2)=√3(f(x+1)+f(x-1)=√3f(x) (iv)
relacing x by x+2 in eq 4...
f(x+4)+f(x)=f(x-2) (v)
adding (iv) and (v) we get
f(x+4)+f(x-2)=0 (vi)
replacing x by x+6 in eq (vi),then
f(x+10)+f(x+4)=0 (vii)
subtracting eq (vi) from eq. (vii),we get
f(x+10)-f(x-2)=0 (viii)
replacing x by x+2 we get..
f(x+12)=f(x)
hence,period of f(x) is 12...
